Site-diluted Ising model in four dimensions

Abstract

In the literature, there are five distinct, fragmented sets of analytic predictions for the scaling behaviour at the phase transition in the random-site Ising model in four dimensions. Here, the scaling relations for logarithmic corrections are used to complete the scaling pictures for each set. A numerical approach is then used to confirm the leading scaling picture coming from these predictions and to discriminate between them at the level of logarithmic corrections.

Revisiting (logarithmic) scaling relations using renormalization group

“…However, the RG flow of the diluted model asymptotically finishes on the line 4u + 3v = O(u 2 ), so we need to include the next (cubic) terms in the perturbative expression and the RG β-functions are no longer quadratic in the couplings. Finally, one finds that u(b) 2 ∼ v(b) 2 ∼ 1/ log b: hence,ϙ = 1/8 as derived in [22,24].…”

Numerical study of the overlap Lee–Yang singularities in the three-dimensional Edwards–Anderson model

“…A fit to a 1 a 2 works well, with a 2 consistent with the value computed from x 1 (a 2 = 1.068(10)). So we have obtained, numerically, a 2 = 1.16(2), at the critical point which has changed to 1.068 (10) in the broken phase.…”

“…Therefore, we take the value of β c = 0.902(8) from [32], which features many more samples and small sizes to control scaling corrections but does not reach the lowtemperature phase, and will also use this reference to check our value of η. Let us first consider a fit of the individual zeros, leaving aside corrections to scaling, i.e., following (10). For the j-th zero, we fit to…”

“…[8,9,10] the analysis of complex singularities was applied successfully to diluted systems (in particular diluted Ising models in two and four dimensions). The key point for the applicability of the standard results, well tested in non-disordered systems, is to compute the complex singularities individually for each disorder realization (called sample) and then use the mean of the individual zeros (sample zeros) in order to test the scaling properties of the zeros and to study the properties of the integrated density of zeros.…”

Numerical study of the overlap Lee-Yang singularities in the three-dimensional Edwards-Anderson model

“…In Refs. [8,9,10] the analysis of complex singularities was applied successfully to diluted systems (in particular diluted Ising models in two and four dimensions). The key point for the applicability of the standard results, well tested in non-disordered systems, is to compute the complex singularities individually for each disorder realization (called sample) and then use the mean of the individual zeros (sample zeros) in order to test the scaling properties of the zeros and to study the properties of the integrated density of zeros.…”